 So I'll get started. So thank you for inviting me, it's a pleasure. I'm sorry I couldn't be there in person, circumstances being as they may. Sure, the weather is much better there than it is here in Chicago. So this work is joint work with John Anuchkun, who's on the faculty at the University of Illinois at Chicago, and this is actually her dissertation topic. We published the paper on which this is based about last September, showed up at Operations Research. And I'll give you just try to step you through the paper and give you some idea of what's in it. It's a pretty long paper. The work started with the discussions we had with GE research labs probably, I don't know, eight, nine years ago, 10 years ago now, something like that. And at that time, GE was very involved, as I suspect they still are, on producing hardware for smart meters around the country. And that was where we originally learned about this topic, smart meters and the ramifications for the power industry, and in particular this question of smart homes and what this new technology can enable in terms of smart homes. So just to motivate things, and I'm sure there are many people in the audience who know more about this particular topic than me, but just to give you some motivation, you may remember or maybe you have no awareness because you live in California of the polar vortex that happened here in Chicago back in January of 2014. These are people social distancing here in downtown Chicago in 2014. And so when this happened, there were all these calls for people to conserve electricity. And the reason was because the authorities were worried that power plants would start to fail because it was negative 40 degrees below zero. And there was concerns about power failures and which would be are highly dangerous in this kind of situation and of course the costs. And so this is the trajectory of the spot prices for electricity in dollars per kilowatt hours over the course of 24 hours on this day. And as you can see, there's a lot of fluctuation with it jumping up to almost $2. So part of the reason why there was all these announcements, these public service announcements to ask the population to conserve electricity is because consumers, as we know, don't see these prices. So the prices that people in the Midwest and Chicago in particular were seeing it's the flat price of 10 cents per kilowatt hour. And so on this day, there was no way to use price signals as a way to cause consumers to curtail their consumption of electricity because all they saw was this red line. And so the hope was by using the radio and television and all that social media that we could somehow get people to curtail. So this, of course, begs the question, what would happen or what would have happened if instead you passed to the consumers that price signal would they have conserved or not? And this is what's known as price-based demand response. So in general, demand response is the idea of managing customer consumption of electricity in response to changing supply conditions. And so this is your old meter that you used to have on your home and it has these little dials and all these dials, I don't know how you read them, but the meter person would come and read the dial and record once a month what your consumption was and it's actually keeping track of cumulative consumption so they take the difference between two readings one month apart and that's how you get your power bill. Nowadays, we've got now these smart meters that are on homes. This is in fact a GE one and these are now electronic. They can record consumption of electricity in very short time intervals of minutes. And so the last time I checked about a third of U.S. homes have smart meters and that number may be higher now. And so that creates an opportunity. So the smart meter becomes now an enabler for all sorts of things. So as I said, there's this communication between the smart meters and common, which is our power comes in, our utility here in terms of relaying power consumption over time. But there's also an opportunity for ComEd to send signals back to the home as well and it could be through the smart meter or it could be through some other internet connection. That happens in various ways. And there's two basic kinds of signals that the power utility can send. One is called direct load, what we'd call direct load control signals which would say, hey, supply is tight, we're having issues. And so you will work the homes, you send out a signal to the homes to conserve electricity and if you've got appliances in the home that can read that signal, they can then downshift into a lower power mode. In some instances, the power utility can actually control air conditioners and things like that and change thermostat settings. So there's different ways in which these direct load control signals can be set up. The other way to do it would be to send dynamic prices. And so rather than try to optimize people's homes and their consumption of power, we instead send them price signals and then we let everybody optimize their consumption relative to those pricing. There's evidence from pilot studies. Each one of these dots is a pilot study. And there's evidence that consumers will respond to price signals. And what I'm showing on the x-axis, this comes from this paper, it's referenced below, looking at the ratio of the peak price to the off peak price. So like the peak price would occur between 4 p.m. and 8 p.m. And so under a peak pricing policy, there'll be a jump in price during that timeframe compared to the off peak times. And so if you charge 20 times, what you normally charge during those peak times, you would expect somebody to start shutting down some lights, in fact they do, and power down their air conditioners or whatever. And furthermore, if you give consumers enabling technologies, you can actually get more benefits. And so here you're seeing reductions of 10 to 20% with price signals only, and you can get more 30%-ish from, if you actually have enabling technologies like light up globes and things like that, alarms and sensors and other technology. So of course this begs the question, how should the prices be set? And there is of course now a large market for these smart thermostats, one of the most famous ones of which is the Nest Learning Thermostat, which many of you, not all of you know about. And the thing about the Nest Thermostat is it's able to respond to occupancy and it's able to learn over time what temperature you like at five o'clock when you're home. And so this is, and it's very easy to use and slick. And so this is usually popular device, and it also has a really nice API in it. To my knowledge, it's not price sensitive. However, it has, as I said, a nice API associated with it where you could program it to be price sensitive. There's another thing about the Nest Thermostat if you actually start looking at the algorithm, it's hard to know exactly what the algorithm is, but just based on what you can learn about how it works, it's occupant aware. So it's aware when someone's home or when someone leaves, people put them in different rooms and it can keep track of whether or not rooms are occupied or not. And so the idea is that when someone leaves the home or when the home is emptied, it can power down the air conditioner because it detects that no one's there. And vice versa, when someone comes home, it can detect that someone's come home and turn on. But to my knowledge, it doesn't anticipate occupancy, which is a different idea. So it doesn't say I'm predicting that my owner will come home with some probability in the next three hours and I'm gonna start to optimize the temperature of my home. It instead is just aware of when people are there or not. And the same thing is true for the prices. It's not, you can program it to be price aware, but basically my understanding is that it's not. Anyway, so the goal of this research, this research has three basic goals. It's to develop a framework for studying demand response based on first principles for doing a few things. So one is just optimizing the price signals of utilities to smart homes to maximize social welfare. And when dynamic pricing in the power industry is discussed, it's often times you hear concerns that the power companies are gonna price gouge and set these prices in some irrational way that profits them. So this is one important question. How do you calculate the right price signals? Two, optimizing the operation of smart homes with price responsive appliances. So if you look at John and Stacy's, she has models for different kinds of appliances in the home and how they would respond to price signals to optimize their operation over time. And then thirdly, to conduct market equilibrium analysis. And this is where sort of the energy policy part of it comes into play. So you wanna be able to answer questions for policy makers like what would happen if you gave everybody a smart meter or smart thermostat and they implemented? What would be the impact on the load, power load and the costs and so on and so forth? That kind of question would be important if you're thinking of subsidizing the equipment and so on and so forth. So in this talk and in this paper, we focus on one and three, and two is in work we've done elsewhere. Quick literature review, this idea of real time or dynamic pricing of electricity has been around for really, really long time. It's been like 50 years at least that economists have stated, oh yeah, customers, consumers should pay marginal costs, should pay spot prices for electricity. So that idea has been around for a long time and it's associated with the peak load pricing literature. In most of this literature, there's an assumption of linear price schedules. It's kind of a soon from the get go. But it's not been until now with all this new hardware, technology, internet and whatnot that we've actually been able to, we actually had the technology to do it. And so it's an exciting area. There's other research that has been done. It seems like it kind of comes and goes over time. So there were people in the 90s that were looking at the technology and people in the 90s that were looking at this idea of having AC controllers with spot prices and there's work now on selling power back to the grid if you've got solar panels and these kinds of things. So anyway, so there's a lot of research and I'm sure I left off, I definitely left off lots of other things but I kind of view it as sort of the economics literature and there's kind of this engineering technology literature. So let me give you just an overview of what the framework that's built in this paper. So our goal in the paper, as I said, was to build a framework that could be used to plug and play. So if you've got your own model for an appliance in the home you could plug it into our framework and study it. In our paper, we look at our version of a smart thermostat which I'll show you in a minute or two. And anyway, but here's the general framework. So we have a power utility and this utility observes some global environmental parameter, WT, which I'm thinking of as the weather but it could be the price of oil and other things. So, but just think of it as the weather and the thing about the weather is that everybody knows the weather. If you look outside, everyone knows the weather. So this is common knowledge to all of the homes. We then index the homes with index I and there's a lot of homes, potentially millions of homes that we're looking at. Each home is able to observe, as I said, the weather, WT, but there's also a home state. So home I in time period T has a state and we're thinking of this state as a multi-dimensional vector of some sort. It could be occupancy and in the case of our smart thermostat, it's whose home, whose not at home, the indoor temperature, those kinds of things. And then what happens then is over time, as T moves from one time period to the next, I'm thinking of time periods of about a day, I'm sorry, of an hour, each home will decide how much load, how much kilowatts to consume for each period and call that QIT and they'll ask the utility to supply that load. And then the key here is that the utility then charges the homes for their consumption. And so what we've done is we've started out rather than just assume as most of the literature does that there's a particular price structure, in particular a linear price structure for how much the utilities are gonna charge for this load for each home. Instead what we've done is started out with the most general function we could and then see if we can say anything about what form it should satisfy. So literally this is just a completely general function PIT, which is the price that is charged by the utility to home I in period T when they consume load QIT and the weather outside is WT or the global environment variable is WT. So now notice that this price function does not depend on the state of the home, which is really crucial to all of this because there's privacy concerns. The power utility can't know who's home and so they can't know the temperature inside your home so on and so forth, at least not by default. So what we're looking for is a price, is a pricing scheme that where there's some asymmetric information where the power utility doesn't know the state of each home, but yet notice that this allows every home to have its own price. So my neighbor could have a different price schedule than me potentially. If they have a big mansion and I have a little small home, maybe there'll be a different price schedule for us. I don't know. All right, so the way the system works is there's a power company and this power company supplies the load which in each period T is the sum of these QITs where I is summing over the homes. So that's the total load supply. And CT is the cost in period, it's a cost function for period T. This cost function is really there to capture, it's a simple function that captures the costs of supplying the power. And it will depend not only on the total load, but also the number of customers that you're supplying to. And it could also depend on WT, which is the global environment variable. Now each home, and I just, one, two, three, all of you up to however many there are, millions, has its own process by which their state, their homes are, the homes are evolving. So each home, so home I has a state H1T which is their state in period one. They then make a low decision, Q1T, and then the power is supplied. They collect some utility. That utility depends on this global variable W, their state of their home and their load requested. And then there's a shock, delta one T, which then dictates how they transition from one state to the next. So then that leads the home into a new state and then they do it again. So the home, each individual home of course, the decisions that, the load decisions that a home makes over time, are there's, the decisions you make in any given period impact the decisions that the home will make later. I mean, if you could serve, if you pre-cool the home, the home will be colder later, so you don't have to run the air conditioner as much. And so each of these homes managing them by themselves, this is a not true real problem. So now, what you could do is you could think about, at least theoretically, setting up an optimization problem that optimizes the whole system. So imagine a power company who had complete visibility into every home and could optimize the load across all the homes. That of course is a crazy problem, intractable, violates basic information, constraints and things like that. So that doesn't make much sense. But what we do, as I said, is we instead have the power company sent each home its own price schedule in the form of these functions P. All right, so the question is, how do you compute the piece? So now, one of the, when we compute the piece, what we're looking at is a special class of policies that are decentralized policies where each agent chooses their own policy that depends only on the information they have available. So the global load W and the state of their home H. And what it means for it to be a decentralized policy is that conditional on the global variable W and the states of all the homes, the probability of seeing a vector of load requests is actually equal to the product of the probabilities of the individual homes. And so that's the class of policies that we're looking at. And so here is the decentralized pricing problem. So what we wanna do is find the price schedules to give to each home such that we maximize the expected social welfare. What is that? Well, that's the overtime, summing overtime. It's the total utility from consuming power minus the cost of supplying that power where each home I is being operated using a policy PII. And that policy depends on the prices that are being sent to that home, the PITs. And so each home solves its own net utility maximization problem. So it's trying to control its air conditioner or some other appliance in the home to maximize the expected utility that the home or the home owners derive from consuming that power minus the price that's paid to central for the power based on this general price function. And of course, there's constraints on the load and there's a transition law that transitions the home from one state to the next. Okay, so that's the problem. Doesn't look easy, but we can say some things about it. So let me give you just an overview of some of the things you can say about it. So for one thing, there exists optimal price functions that satisfy this equilibrium condition, which looks complicated, but intuitively it says that the price, there exists an optimal price function such that the price that's given to home I in time period T for consuming load QIT given the weather's WT is the expected incremental cost of adding this home I's load to the whole city. So this home load is this little QIT. So this is the cost of adding this home's load to the whole city minus the cost of just pretending the customer doesn't exist at all. So this is the incremental, it's the expected incremental cost to the system. So that part of it is straightforward, but this thing is a lot more subtle than just that. Because it's an equilibrium condition. So and so what's happening is that the prices depend on the policies that the customers are using, the homes are using it to decide their loads. On the other hand, the policies that the customers are using depend on the prices. They show up here. So prices are on both sides, the left side and the right side. Okay, so it's an equilibrium condition. And the homes themselves, even though it looks as if the homes are kind of not connected, they are connected to each other because the problem is that if I don't consume power now and just let my house warm up in the middle of summer, then in another hour or two, I might chill it. And then that could cause the prices, the load on the system, the central system generate the generation system to go up, which then causes the price of my neighbor to go up. And my neighbor is responding to that price. And the solution to this is an equilibrium of all of those inter-temporal complicated dynamics that occur. Maybe my neighbor's house isn't very well insulated, mine is and all of that's gonna come into play here in calculating these prices. So an important question to ask is when are optimal price schedules linear? Because so much of literature just assumes they're linear, it's a useful question to ask, why doesn't a linear price schedule just work for you? Well, here is how our problem simplifies in a very simple case of one period only, there's no weather outside, there's only one agent. And so in this case, what we're doing is we're just trying to find a price function P, P of Q, that maximizes the expected social welfare, there's only one agent, subject to for every state of the home, the agent will choose a load that maximizes its utility. Okay, so this is analogous to our situation much simplified because the P here only depends on the load and not the state of the home, okay? Now the key is that if you allow the price to depend on both the state of the home and the load, then it's not too hard to see that it's optimal to set the price schedule to be linear in Q. So the marginal cost is to be evaluated at the optimal load Q star of H, okay, which doesn't depend on P. And so it's just that number depends on the state of the home different for each state of the home, times the Q. And if you plug this in, you see in fact that this works and it does solve the social welfare problem. Okay, but this is not our setting because you can't observe, we can't have the prices be dependent on the state of the home. So if you stare at this enough, you realize that, well, one way to solve this problem is just to let P equal to C. So if you literally just pass the whole cost function down to the agent, then they'll do the right thing to maximize social welfare, but then that cost function is gonna be nonlinear. Not linear, okay? So we have some analysis in the paper. This is kind of technical, but basically what it says is that if you're adopting an optimal decision rule that the price schedule that induces that is it has sort of the same shape as the price function around the optimal loads that you choose. So all the derivatives are equal just kind of locally around. So what that means is that you're not in general gonna get a linear price schedule out of this thing unless the cost function is linear. But our cost function is not linear. In fact, if you look at the cost curves, generation cost curves based on different sources of electricity, the costs go up, their convex non-decreasing in the load. So what we do is we set up a cost function which is for the whole city. So QIT here denotes a city of agents or homes I here. So T and we're summing up over all of the homes. So this is the total load. And so the cost is a function of the total load, the number of homes in the city and the weather outside we assume that it takes this form. So this form is I, so it's a number of homes times the cost per home. And the cost per home is based on some cost function which is a function of the average load per home and the weather outside. And so the idea here is that as a city gets large, the cost to supplying power to a home doesn't go to zero. There's some number, but asymptotically it grows assuming that the average load of a home is converging to something, then the cost looks sort of linear asymptotically in the number of homes. All right, but the cost function itself, the little C here is convex. Now, if you assume this and it's a reasonable assumption under some other basic math assumptions including that you get some kind of strong law of large numbers kind of thing, then it turns out that the optimal price signal to send in fact is just a P times Q, which is the assumption we often just make. It's P times Q where the price depends on time. It depends on the weather, WT. It does not depend on the home. So homes get the same linear price signal. And we call this the small home result because even if my neighbor has a McMansion he's still small compared to the whole city. So, and I'm small too. So, and so that's why it's us, we call this the small home result. All right, so that's really, really useful. One of the questions, one of the audience has a question on the modeling that may be nice if you want to answer. Yeah. It's not the Q and A. Sure, I can't, for some reason I can't see that little panel, let me see here. Right. Yeah, it doesn't show up. So. Yeah, do you want to, why don't you admit them because I don't seem to be able to? Nikola, would you mind raising your hand and I can unmute you to ask your question? Nikola, if you hear, can you raise your hand and I can unmute you for asking your question? So, the question that has been asked here. So, here she is. Okay, you can ask your question now. Yeah, so my question was, is there a way to integrate learnings around individual home cooling behavior? For example, it doesn't have to be cooling, but that you can integrate into the model to be a predictor of demand. Yeah, yeah, yeah. Yeah, that's a great question. I'm actually suppressing something important here. So, let me see. So back right here. So you see Pi I is in the arc max of this thing. So, what's happening is, is that every home has its own mark off decision process, essentially that it's solving. And inside of this, this home is evolving over time according to its own thermodynamic properties. So its own heat transfer, its own insulation, all that kind of stuff. So, that's all in here. And when we go and compute numbers with it, as I'm about to show you, we do take into account all of those different home specifications. So, for example, you could figure out what the impact would be of changing, if you could double everyone's insulation across the city, what would be the impact on the price in a market equilibrium, we could totally do that. But also, as you were mentioning, Nest, for example, is learning about the individual home. And when people are home, maybe it's cycling of the cooling, that sort of thing, that you might integrate into the model. Yeah, yeah, yeah. So this is, so I'm assuming underneath the hood that we've learned the transition probabilities for occupancy, say, and the parameters that dictate the utility function, how hot or cold you like it in your house. So I'm assuming we've already collected enough information to be able to populate these home specific functions, if that makes sense. So yeah, but you could, if you didn't know them, and you had to learn them over time, that would be an interesting extension of all this. Great, thank you. Yeah, sure. Yeah, that's a good question. Where am I? Yeah, all right. So let me see if I can try to, let me try to just wrap up because we're to leave some time for questions. But we have an algorithm for adapting these prices and for simulating a whole city, what you can do is you can partition the homes into different types of homes. And then for each home type, you can aggregate them together to simulate a whole city. And this is kind of an adaptive algorithm for tuning those prices over time. So even though we're not learning about the homes, preferences and whatnot, we are learning the prices over time here in order to compute the equilibrium. Let me just say something about the thermostat. So I'm gonna show you some numerical results from running this, computing these prices and whatnot. And I do it on our version, our vision of the smart thermostat, which is quite a bit smarter than Nest, not on the front that Nicola just brought up, but in terms of anticipating occupancy and anticipating price movements. And so we're not only gonna learn, so occupant aware means not on the learns the behavior, but anticipates it. So it's anticipating that the owner is gonna come home at five o'clock with some probability and with some probability they don't come home till midnight and it's price responsive, meaning that it sees that the price, it's anticipating that the price is gonna go up by early afternoon and we better do something about it now. I'm not gonna go into detail about how we fit, here's a dynamic program that does all this for the smart thermostat. Let me just show you, now that you have some sense of what our thermostat does, the key is that it moves the, it consumes power for the air conditioner based on anticipating price movements and occupancy. That's the main thing you need to know. So here I've got to, just to give you an illustration, got one home, it's occupied by a single person who lives alone and they have a very regular schedule and there's their schedules every day is the same and they sleep, they get a good eight hours of sleep, so this isn't a student and they're active early in the morning for a couple hours. The hottest day of 2011 was July 20th and so I'm just gonna show you what happens on that day and they're willing to pay 20 cents. So per kilowatt hour. So here is the outside temperature on that day. This is the hottest day of the year. It's spiked up to almost 100 degrees on that day in black is what you see. The prices on that day, the actual prices were in green, these are the spot prices coming from the market and then the flat price is this 10 cents per kilowatt hour, okay, to give you sense. So this is one of the main things I want to show you is that this, when you anticipate things, you get interesting behavior. So the green and the red are just, these are ignoring, the red is just ignoring everything. It's just a flat price. It doesn't keep track of who's home. The green is a little bit smarter, but the main thing is that I want to show you is the blue and the purple. So in the blue, this is a flat price. So there's no price anticipation at all. It's just 10 cents on this day, 10 cents per kilowatt hour, but it is occupant to wear. And so what it's doing here is, this is the indoor temperature over time. Our homeowner likes it like 40, 74, 75 degrees, something like that. And as you see at around, it's basically when the homeowner goes to work or school, the home heats up to almost 80 degrees here, 78 degrees. And then at like one or two o'clock, it starts to cool it off because it's anticipating that the homeowner is gonna come home and they don't like it that hot. They like it a little cooler. So it's anticipating and operating by itself, even though no one's home. In contrast, if once you put the prices in there and you allow it to anticipate the prices, it actually has exactly the opposite effect. So the homeowner leaves at nine in the morning and look what it does. It turns the home into an ice cube. I mean, it's down to 67 degrees at noon and no one's home. So why is it doing that? It's doing that because it's anticipating that the price is gonna be extremely high later on in the day. And so it's pre-cooling the home so it can shut things down and let the home warm up so that in anticipation of the homeowner coming home at five, it's a good 74 degrees, which is how they like it when they come home. And so you get very different behavior here. All right, so just quickly, we ran the whole combat service region 3.4 million residents in July of 2011. Data from all sorts of places. We fit the cost function using a function from the literature that we call the Barlow, that's called the Barlow function. We compare baseline, which is what we do now. Nothing smart, maintain a constant indoor temperature, flat pricing with what happens when you put any smart thermostats under flat pricing or peak pricing where you double the price during peak hours versus our prices that come out of our model. So this is the flat pricing. And I'm showing this as a function of the saturation. So as a 100% one means everyone has a smart thermostat and 0.01 means basically very few people have it. So you can see how the curves change. This is not so interesting. The main thing is that my thermostat knows you're sleeping and so it raises the temperature to conserve power and that's why the load is a little lower at night for me. But this is interesting. So if, and this is something you don't see in practice. So if you do peak pricing, so you double the price between 4 p.m. and 8 p.m., my thermostat, which anticipates these price movements, what it does is it consumes a ton of load just before the price increase. Why? Because it wants to cool the homes or our homeowners comfortable during this period, right? So it gets it while it's cheap and so you get a big peak before the load and then it kind of shuts things down and then right after the peak, it then turns back on again. So there's a lot of concern about there being a second peak by doing this dynamic pricing, creating it just moving the peak around. What this is showing is that peak load pricing that can actually create two peaks. And if you don't see this in practice, it's because people don't have thermostats that are as smart as mine. Anyway, so there's that. And then there's dynamic equilibrium prices. These are the prices actually. I'm showing prices on the Y axis over the course of this day. And as you see, they do go up. This is this hot peak summer day. As saturation, the saturation of smart thermostats goes up, you see that the prices drop because there's positive externalities to having all these smart meters installed. And then this is the load. And what you see is this behavior of pre-cooling in the morning versus in the later on the day conserving. That's kind of the behavior that you get. And it's more accentuated the more smart meters you've got or smart thermostats you've got. So anyway, this is my last slide. So just a summary. So we have a generalizable pricing methodology for infinitesimal agents, which are these smart homes. I've shown you that the behavior of our smart thermostat, it pre-cools in the morning under dynamic pricing. Whereas flat pricing will warm in the morning, very different. I showed you that peak pricing can cause two different load peaks. In terms of the social welfare numbers, I've just kind of summarized here because there's a lot of numbers in the paper. But the dynamic equilibrium pricing reduces the monthly power bills by as much as 41% based on the assumptions, calculations in our paper, which is pretty good versus 19%, which is closer to what like the nest does basically. You increase social welfare, you reduce the monthly generation costs, you reduce the system-wide peak load. So all these great benefits of using smart thermostats and doing dynamic pricing. But, and I'll end here, what we see is that there's a decrease in monthly supplier surplus by as much as 50% when you do this dynamic pricing. And so I would assert that this offers some explanation for why we see such a slow adoption of dynamic pricing, real-time spot pricing in the consumer, the residential power side of things, because power companies make money on selling electrons. And so they sell fewer electrons, less revenue coming in. And anyway, it looks like they actually, their surplus would actually go down significantly. So in order to get all these benefits, this suggests that there would need to be some government intervention of some sort around regulation, supporting the industry to make this more wide, something that happens more widely. So anyway, so I'll just leave it at that. And I'll take any questions. Thank you. Thanks, Professor Edelman. It was a really nice presentation, especially now that everyone is at home. We do need more of these smart metering at homes. We have a few questions. I would like to ask people who have submitted their question to Q&A to raise their hand and ask it orally from Professor Edelman. So John, you are on. Go ahead. John, you can ask your question. Okay, hi. The idea of cooling the house when no one's in it is using the thermal mass of the house as storage, basically. But it's quite possible to store energy in forms that wouldn't directly affect the temperature of somebody's in the house. I mean, you wouldn't do this trick if somebody's in the house during the day. You wouldn't say, oh, we'll just chill you down to 48 degrees because you will save money later. So where do you see this kind of dynamic pricing if more homes had thermal storage? For example, you could make ice in a storage system when power is cheap, whether it's the middle of the night or the afternoon or whatever. And then you use this thermal reservoir to cool the house. You can also do this for heat, storing energy essentially in a phase transition material and then using that to heat the house. I'm curious if you've thought a little bit about how heat storage in a house would change your pricing models. Yeah, yeah, that's very interesting. So there's different kinds of power consumption. There's the storeable stuff. There's the pure discretionary. And then there's the load that can be the fur, right? Timeshifted somehow. So, well, first of all, I would say, yes, those are all great ideas. I think in terms, we haven't analyzed how these different variations of storeable energy would change the pricing, but you certainly could do that with the framework. So right now, the model has a heat transfer model that keeps tracking, that has some notion of the thermal mass of the home and that you're right, it's a storage device. If there was a way, so the model currently as it's implemented in the paper, if it would cool the home and make people, it is taken into account the probability that someone comes home for lunch, it's freaking freezing in the home, right? And so it does do that. If you had instead, you just substitute your model that has a thermal mass that doesn't, you know, pinch on the consumer in an instead, then my guess is that it would end up actually storing even more of the energy in the thermal mass, you know, with the thermal mass. And what, in fact, would it have on the price? It'd probably be beneficial would be my guess, but you could certainly just adjust, come up with the model of how the home chooses to consume load to build up that energy mass over time, plug it into the framework and then investigate a question like that. That'd be very interesting. Hi, Dan. So another question in terms of with the, using the price signal to shift behavior, do you run the risk of creating like the Google map problem whereby if every smart thermostat is doing the same thing that you just end up shifting where the peak load or the expensive loads occur? Yeah, yeah, yeah. Yeah, so when we look at the equilibrium, we don't see that, right? That's a concern that's frequently brought up, that everyone will just shift all their load to some other time and they would all coordinate on that same time. But there's heterogeneity in the homes, as we were saying earlier, and there's the heterogeneity comes into play not only in terms of the thermal properties of the homes, but also the utility functions, their willingness to pay for power and also in the probabilities, the occupancy probabilities that evolve over time, that like the Nest thermostats learning and whatnot. So in our experiments, it doesn't, it looks like there's enough heterogeneity in randomness that that behavior where everybody just flips everything on at eight o'clock doesn't happen. Oh, interesting, that's great. I mean, if you had a deterministic model where everybody looked the same, maybe you'd see some of that, but when you've got probability of homeowner coming home at five o'clock is 0.5 versus 0.3, the neighbor's out, you just don't see that. Great, I had a quick relevant question and it's about this implementing this in a dynamic way and adaptive system, that assuming the Nest would like to learn the patterns that the household has and take over and make quick decisions that the household cannot do in real time. Is there possibility that because of the noise, the learning algorithm becomes unstable and eventually does unstable control of the house temperature? Yeah, so as I said earlier, I mean, our model just assumes that you know those numbers, you've learned those numbers and they're not changing, you know, over time in a way that we don't already understand. So we haven't looked at what happens when you try to learn on top of this, what those parameters are. You know, my intuition about the learning problem is that learning is pretty quick. I mean, you know, if I was Nest and I had data on lots of millions of people using my thermostat, I probably could guess your parameters pretty well with a little survey. I don't know if they do that or not, but I would think you could probably guess if you give me a million customers that have used my Nest thermostat for years and you tell me some basic demographic information, information about the size of their home, how many people are there, you probably could come up with reasonably good parameters. So, you know, is the thing really learning anymore? I mean, does the darn thing learn? I mean, it seems like after installing it a million times, there's really not much more learning. Maybe I like it 72.6 and you like it 72.4, but what's the value of that learning? I don't know, you know, maybe my schedule is different. You know, once you tell me I'm a third shift worker, well, that changes, you know, well, they probably have a pattern for that somewhere. So I'm not sure how interesting the learning problem is, is what I'm suggesting. And even if you did, even if you had a learning, I think you could learn quickly with some limited data. In fact, I mean, even from like a thermodynamic perspective, like all of the heat transfer coefficients and whatnot, with enough data from a single home, you can include, I think, what, you know, what those parameters are, probably fit them reasonably well. I haven't done that exercise, but I'm guessing with enough data from a single home, you could do that. So I don't know. And it's probably not even doing anything that smart, by the way, I don't think it's, I don't think there's a heat transfer model inside the Nest thermostat. I could be wrong, but I don't think there is. I think they're just learning what temperature you like. Thank you, Professor Adelman. You're at the end of the time for this seminar. Thanks again for the time you spend and the presentation. You're welcome. I hope it was helpful and interesting. Thank you all. Bye.